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Recurrent sequence with divisor function is unbounded, but not monotone

Source: VJIMC 2024, Category II, Problem 4

April 14, 2024

functionDivisorsSequencesIntegersnumber theory

Problem Statement

Let

(b_n)_{n \ge 0}

be a sequence of positive integers satisfying

b_n=d\left(\sum_{i=0}^{n-1} b_k\right)

for all

n \ge 1

. (By

d(m)

we denote the number of positive divisors of

m

.) a) Prove that

(b_n)_{n \ge 0}

is unbounded. b) Prove that there are infinitely many

n

such that

b_n>b_{n+1}